On some tensors of six-dimensional Hermitian planar submanifolds of Cayley algebra
Abstract
In the present note, we consider six-dimensional Hermitian planar submanifolds of Cayley algebra. The almost Hermitian structure on such a six-dimensional submanifold is induced by means of so-called Brown — Gray three-fold vector cross products in Cayley algebra. The six-dimensional Hermitian planar submanifolds of the octave algebra contain all six-dimensional Kählerian submanifolds of Cayley algebra. However, there exist non-Kählerian six-dimensional Hermitian planar submanifolds in the octave algebra.
The components of the tensor of the Riemannian curvature for a six-dimensional almost Hermitian planar submanifold of Cayley algebra are computed. Remark that the tensor of Riemannian curvature plays a fundamental role in geometry of almost Hermitian manifolds. Knowing all components of the tensor of the Riemannian curvature for a six-dimensional almost Hermitian planar submanifold of the octave algebra, it is possible to study so-called Gray’s identities for this submanifold.
The components of the Ricci tensor and of the tensor of conformal curvature (known also as Weyl tensor) for a six-dimensional almost Hermitian planar submanifold of Cayley algebra are also computed.
Reference
1. Kirichenko, V. F.: Classification of Kählerian structures, defined by means of three-fold vector cross products on six-dimensional submanifolds of Cayley algebra. Izvestia Vuzov. Math., 8, 32—38 (1980).
2. Banaru, M. B.: Geometry of 6-dimensional Hermitian manifolds of the octave algebra. J. Math. Sci. (New York), 207:3, 354—388 (2015).
3. Gray, A.: Six-dimensional almost complex manifolds defined by means of three-fold vector cross products. Tôhoku Math. J., 21:4, 614—620 (1969).
4. Banaru, M. B., Kirichenko, V. F.: The Hermitian geometry of the 6-dimensional submanifolds of Cayley algebra. Russian Math. Surveys, 49:1, 223—225 (1994).
5. Banaru, M. B., Banaru, G. A.: A note on six-dimensional planar Hermitian submanifolds of Cayley algebra. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, 1 (74), 23—32 (2014).
6. Banaru, M. B., Banaru, G. A.: 1-cosymplectic hypersurfaces axiom and six-dimensional planar Hermitian submanifolds of the Octonian. SUT J. Math., 51:1, 1—9 (2015).
7. Banaru M. B., Banaru G. A.: On planar 6-dimensional Hermitian submanifolds of Cayley algebra. DGMF, 48, 21—25 (2017).
8. Gray, A.: Curvature identities for Hermitian and almost Hermitian manifolds. Tôhoku Math. J., 28:4, 601—612 (1976).
9. Kirichenko, V. F.: Differential-geometric structures on manifolds. Odessa (2013).